#195406

order	512
nilpotency class	3
isoclinism	isoclinic class of #18446
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	#60940	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1166 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,5,8,6,10,4,9)(3,11,12,16,14,15,13,7)(17,18)(19,28,32,39)(20,33)(21,26)(22,37)(23,27)(24,31,40,30)(25,35)(29,44,47,38)(34,42)(36,41)(43,45,48,46);(1,3)(2,7)(4,12)(5,13)(6,14)(8,15)(9,11)(10,16)(17,19,22,31,23,32,21,30)(18,24,37,28,27,40,26,39)(20,29,34,45,36,47,35,46)(25,38,33,43,42,44,41,48);(2,8)(4,5)(7,15)(9,10)(11,16)(12,13)(17,20)(18,25)(19,29)(21,35)(22,34)(23,36)(24,38)(26,41)(27,42)(28,43)(30,46)(31,45)(32,47)(33,37)(39,48)(40,44);(1,4,6,5)(2,8,10,9)(3,12,14,13)(7,15,16,11)(17,21,23,22)(18,26,27,37)(19,30,32,31)(20,34,36,35)(24,39,40,28)(25,33,42,41)(29,45,47,46)(38,43,44,48)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fhi,e^b=eh,f^b=fi,g^b=gh,d^c=dh,e^c=ei,f^c=fh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 102602);